Properties

Label 2-2366-1.1-c1-0-4
Degree $2$
Conductor $2366$
Sign $1$
Analytic cond. $18.8926$
Root an. cond. $4.34656$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 2·5-s + 7-s − 8-s − 3·9-s + 2·10-s − 4·11-s − 14-s + 16-s − 6·17-s + 3·18-s − 2·20-s + 4·22-s + 8·23-s − 25-s + 28-s − 10·29-s + 8·31-s − 32-s + 6·34-s − 2·35-s − 3·36-s − 6·37-s + 2·40-s + 6·41-s + 4·43-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.377·7-s − 0.353·8-s − 9-s + 0.632·10-s − 1.20·11-s − 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.707·18-s − 0.447·20-s + 0.852·22-s + 1.66·23-s − 1/5·25-s + 0.188·28-s − 1.85·29-s + 1.43·31-s − 0.176·32-s + 1.02·34-s − 0.338·35-s − 1/2·36-s − 0.986·37-s + 0.316·40-s + 0.937·41-s + 0.609·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2366\)    =    \(2 \cdot 7 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(18.8926\)
Root analytic conductor: \(4.34656\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2366,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5981342555\)
\(L(\frac12)\) \(\approx\) \(0.5981342555\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
7 \( 1 - T \)
13 \( 1 \)
good3 \( 1 + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.791683393935935773346976116547, −8.310993556895860538285090281250, −7.55111798933411457737769636175, −6.98546602109746736864670767573, −5.85357656265532644668678830790, −5.09668062546972176782846052974, −4.10388224753287373041649125168, −2.96897904490708863142085706195, −2.21452719189805902343262068063, −0.52137795477594883297470319142, 0.52137795477594883297470319142, 2.21452719189805902343262068063, 2.96897904490708863142085706195, 4.10388224753287373041649125168, 5.09668062546972176782846052974, 5.85357656265532644668678830790, 6.98546602109746736864670767573, 7.55111798933411457737769636175, 8.310993556895860538285090281250, 8.791683393935935773346976116547

Graph of the $Z$-function along the critical line